Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

Two Regular Polygons

Stage: 4 Challenge Level:

Two polygons fit together so that the exterior (orange) angle at
each end of their shared side is $81^\circ$.
If both shapes now have to be regular polygons, but do not need to
be the same, and each polygon can have any number of sides, could
the orange angle still be $81^\circ$, and if that is possible how
many sides would each polygon have?
Find solutions for when the orange angle is $27^\circ$ and when it
is $54^\circ$.

Can you make a conjecture about the connection between the size of
the orange angle and the number of sides on each polygon.
If you can, are you able to justify your conjecture?

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